Matrix Algebra                                   Matrix Addition, Scalar Multiplication and Transposition 
                           2 
                           Matrix Algebra 
                            
                            
                            
                           Section 2.1. Matrix Addition, Scalar Multiplication 
                                                and Transposition 
                            
                                                            A  rectangular  array  of  numbers  is  called  a  matrix  (  the  plural  is 
                                                matrices ) and the numbers are  called entries of  the  matrix. Matrices are 
                                                usually denoted by uppercase letters:  A,B,C and so on. Hence, 
                                                                   1 2         1                    1      1               1
                                                            A     0 5 6                      B      0     2   C            3  
                                                                                                                             1
                                                are matrices. Clearly, matrices come in various shape depending on the 
                                                number of rows and columns. For example, the matrix A shown has 2 rows 
                                                and 3 columns.  In general, a matrix with m  rows and n columns is referred 
                                                to as an  m n matrix or as having size  m n. Thus matrices  A,B,C above 
                                                have sizes  2 3,2 2,3 1, respectively. A matrix of size  1 nis called a 
                                                row matrix, whereas one of size m 1 is called a column matrix.  
                                                            Each entry of a matrix is identified by the row and column in which it 
                                                lies.  The  rows  are  numbered  from  the  top  down,  and  the  columns  are 
                                                numbered from left to right. Then the  (i, j)- entry of a matrix is the number 
                                                lying simultaneously in row i and column  j. For example: 
                                                            The (1,2)entry of   A             1 2         1  is 2 
                                                                                              0 5 6
                                                                                                                                                                 
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                                 Matrix Algebra                                   Matrix Addition, Scalar Multiplication and Transposition 
                                                                                                     The (1,2)-entry of B                       1        1  is        1 
                                                                                                                                               0       2
                                                            A special notation has been devised for the entries of a matrix. If  A is an 
                                                            m n matrix, and if the  (i, j)- entry is denoted as  aij , then  A is displayed 
                                                           as follows: 
                                                                                                 a         a         ...     a
                                                                                                   11        12                1n
                                                                                        A        a21       a22       ...    a2n  
                                                                                                  ...        ...     ...      ...
                                                                                                am1 am2 ... amn
                                                           This is  usually denoted simplify as  A                                        a . An  n n is called a square 
                                                                                                                                            ij
                                                           matrix . For a square matrix, the entries :  a ,a                                             ,..., a      are said to lie on 
                                                                                                                                              11      22         nn
                                                           the main diagonal  of the matrix.  
                                                            
                                                           Two matrices  A and B are called equal ( written A =B ) if and only if : 
                                                           1.            They have the same size 
                                                           2.            Corresponding entries are equal 
                                                                         or  can  be written as  a                          b  means that a                       b  for all i, j .  
                                                                                                                    ij        ij                            ij      ij
                                  
                                 Example 11  Given A                             a     b ,        B        1 2            1  ,  C            1        1 ,  discuss  the  possibility 
                                                                                 c     d                   0 5 6                            0       2
                                                           that  A         B, B C, A C 
                                                           Solution: 
                                                            A  =B is impossible, because  A and  B are of different sizes. Similarly, 
                                                            B C is impossible.  A C is possible provided that corresponding entries 
                                                           are equal:  a                b = 1             1  means a 1, b                           1,    c 0, d 2.  
                                                                                  c     d        0       2
                                                            
                                                                                                                                                                                                       
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                           Matrix Algebra                                   Matrix Addition, Scalar Multiplication and Transposition 
                                                Matrix Addition 
                                                If  A and  B are matrices of the same size, their sum  A                                  B is the matrix 
                                                formed by adding corresponding entries. If  A                             a  and  B          b , this take 
                                                                                                                           ij                  ij
                                                the form: 
                                                                       A B a b  
                                                                                     ij     ij
                                                 Note that addition is not defined for matrices of different sizes. 
                            
                           Example 12  If   A               1      1       1  and B          1 2         1 , compute  A            B! 
                                                            3      2     4                   0 5 6
                                                Solution 
                                                 A B= 1 1                 1 2         1 1         2 1          2  
                                                             3 0          2 5 4 6                 3 3 10
                            
                           Example 13  Find a,b,c if  a b c                           c   a b         3 2         1  
                                                Solution 
                                                Add the matrices on the left side to obtain: 
                                                  a c b a c b                    3 2         1  
                                                Because  the  corresponding  entries  must  be  equal,  this  gives  three 
                                                equations: 
                                                 a c 3,b a 2,c b                      1. Solving these yields a               3,b       1,c    0. 
                                                 
                                                The properties of  Matrix  Addition 
                                                If  A,B,C are any matrices of the same size, then: 
                                                1.  A     B B A                                         ( commutative law ) 
                                                2.  A     (B C) (A B) C                                 ( Associative law ) 
                                                 
                                                The m n matrix in which every entry is zero is called the zero matrix and is 
                                                denoted as 0, hence,  
                                                                                                                                                                 
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                           Matrix Algebra                                   Matrix Addition, Scalar Multiplication and Transposition 
                                                3.  0      X X 
                                                The negative of an m n matrix  A ( written as  - A ) is defined to be m n 
                                                matrix  obtained  by  multiply  each  entry  of  A  by                             1.  If    A       a ,  this 
                                                                                                                                                      ij
                                                becomes              A       a , hence, 
                                                                               ij
                                                4.     A ( A) 0 
                                                for all matrices  A           a  where 0 is the zero matrix of the same size as  A.  
                                                                               ij
                                                A closely related notion is that of subtracting matrices. If  A,B  are two m                                   n 
                                                matrices, their difference  A               B is defined by: 
                                                 A B A ( B), i.e. : A B                        a       b       a      b  
                                                                                                ij      ij      ij     ij
                                                 
                           Example 14  A                   2 1 ,   B          3       2 ,  C        1 1  
                                                         0     1              2     1               2 2
                                                 Compute  A,A B,A B C 
                                                Solution 
                                                    A      2      1  
                                                           0      1
                                                 
                                                 A B=            2 3 1 2                  5 3  
                                                               0    2     1 1             2 0
                                                 A B C                2 3 1 1 2 1                    0       2  
                                                                    0    2 2 1 1 2                   0     0
                            
                           Example 15  Solve  3                    2 +X= 1 0 , where X is a matrix. 
                                                              1 1                  1 2
                                                Solution 1 
                                                 X must be a 2 2 matrix. If  X =  x                      y , the equation reads: 
                                                                                                    u    v
                                                                                                                                                                 
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